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For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.   


Characterizations; Completely Monotonic Functions; Order Statistics; Record Values; Truncated Moment.

Article Details

Author Biographies

Mohammad Shakil, Miami Dade College, Hialeah Campus

Dr. M. Shakil, Ph.D., CStat (RSS, UK), PStat (ASA, USA), Member - AMS, ILAS & RGMIA
Professor of Mathematics, Department of Liberal Arts and Sciences - Mathematics
Miami Dade College, Hialeah Campus, 1780 West 49th Street, Suite 2325
Hialeah, Fl. 33012, USA
Telephone: (305) 237-8805 (Office); e-mail:
Editor-in-Chief: POLYGON (MDC, Hialeah Campus, Florida, USA)
Member of the Editorial Board: Jñānābha, JSTA, IJASP, PJSOR; Reviewer: Mathematical Reviews (AMS).

Dr. Shakil is teaching as a Full-time Faculty in the Mathematics Dept. of Miami-Dade College, Hialeah Campus. Prior to joining MDC, Dr. Shakil taught at Florida International University, Nova Southeastern University, Johnson & Wales University, Broward Community College and several institutions in India and Nigeria. He holds a master's and a doctoral degree in mathematics from Patna University in India, and a master's degree in statistics from Florida International University. His research interests lie in the fields of numerical and functional analysis, statistics and probability, history of mathematics, mathematics and statistics education and curriculum development. He has several research publications to his credit in the field of mathematics, statistics and education.

Dr. Mohammad Ahsanullah, Rider University, Lawrenceville, New Jersey, USA

Professor Emeritus, Dept. of Management sciences, Rider University, Lawrenceville, New Jersey, USA

Dr. B. M. G. Kibria Kibria, Florida International University, Miami, Florida, USA

Professor, Dept. of Mathematical Sciences
Florida International University, Miami, Florida, USA

How to Cite
Shakil, M., Dr. Mohammad Ahsanullah, & Kibria, D. B. M. G. K. (2021). Characterizations of Some Probability Distributions with Completely Monotonic Density Functions . Pakistan Journal of Statistics and Operation Research, 17(1), 51-64.


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